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Gravitational Lensing
Gabriel Bartosch Caminha([email protected])
Special lecture for the course “Physics of Galaxies”; May-2020
Some publicly available references:● M. Meneghetti notes
• www.ita.uni-heidelberg.de/~massimo/sub/Lectures/gl_all.pdf● J. Wambsganss, 1998, Gravitational Lensing in Astronomy, Living Reviews in Relativity
• https://arxiv.org/abs/astro-ph/9812021● C.S. Kochanek, 2004, The Saas Fee Lectures on Strong Gravitational Lensing
• https://arxiv.org/abs/astro-ph/0407232● M. Bartelmann, 2010, Gravitational Lensing
• https://arxiv.org/abs/1010.3829
Some books:● Schneider, Ehlers & Falco, 1992, Gravitational Lenses, ISBN 9783540665069● Mollerach & Roulet, 2001, Gravitational Lensing And Microlensing, ISBN 9789810248529● Petters, Levine & Wambsganss, 2001, Singularity Theory and Gravitational Lensing,
ISBN 9780817636685● Congdon & Keeton, 2018, Principles of Gravitational Lensing: Light Deflection as a Probe of Astrophysics
and Cosmology, ISBN 9783030021214
2Outline
General concepts (2/3 of the time)➔ A little of history➔ First observations➔ Lens equation and potential➔ Magnification and distortion➔ Gravitational lensing by stars, microlensing➔ Gravitational lensing by galaxies and galaxy clusters
➔ Strong lensing➔ Weak lensing
Applications (1/3 of the time)➔ Mass distribution of galaxies and galaxy clusters➔ Large scale structure of the Universe➔ The distant and faint Universe
3What is gravitational lensing?The deflection of the light of a background source (galaxy) by a massive object
Credit: NASA / ESA
Credit: NASA / ESA
Lens by a single galaxy
It is possible to emulate some gravitational lensing effect with optical lenses For example the base of a wine glass
Credit: https://www.um.edu.mt/think/the-einstein-enigma/
4What is gravitational lensing?The deflection of the light of a background source (galaxy) by a massive object
Lens by a massive galaxy cluster, Abell 370
Credit: NASA / ESACredit: NASA / ESA
Lens by a single galaxy
5History
1783 – Letters between John Mitchell and Henry Cavendish discussing about the effects of a massive star on the light.
➔ First attempt to compute the light deflection by a gravitational field using Newtonian physics, wrong value
1919 – Arthur Eddington and his collaborators observations of the light deflection of distant stars by the Sun during an total eclipse.
➔ The most notorious test of general relativity was performed using gravitationallensing (May 29 1919)
1924 – Orest Chwolson (Astronomische Nachrichten, 1924, 221, 329) multiple images of background stars could be formed. Moreover, with a perfect alignment a image of a ring will be created.
1936 – Einstein makes his own calculations (Science, 1936, 84, 56), and in addition finds that the apparent brightness of a background star can be magnified by the “lens-like action of the gravitational field”.
1987-89 – Discovery of elongated, curved features around galaxy clusters. First “detection” of gravitational arcs (Soucail+1987 and Petrosian+1986)
6History
Soucail+1987 (CFH Telescope)
HSTHubble Frontier Fields
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strong regime● creation of multiple images● gravitational arcs● high magnification (10-100
times the intrinsic lum.)
weak regime
● weak regime● small distortion of
the images of the galaxies
credit: NASA/ESAAllow us to observe very faint and distant object
Magnifications preserves surface brightness:
Gravitational lensing
Deflection of the light of a background galaxy by a massive object: galaxies, galaxy clusters, stars
Measure the mass distribution of these systems● Does not depend on any
assumption of the dynamical state or baryonic processes of the object acting as lens
Detect/measure substructures of the Dark Matter distribution
8The content of the Universe
Stars, 0.5%Free Hydrogen and Helium 4%Neutrinos and heavy elements ~0.1%
We know little about Dark Matter, and even less about Dark Energy
Credit: NASA / WMAP Science Team
With gravitational lensing we can probe the dark sector of the Universe
9The Bullet ClusterMass distribution of a massive galaxy clusterImage: visible light, galaxies; Red: Hot gass; Blue: Dark MatterDark Matter does not interacts directly with baryons and light, only gravitationally
Credits: X-ray: NASA/CXC/CfA/M.Markevitch et al.;Optical: NASA/STScI; Magellan/U.Arizona/D.Clowe et al.;Lensing Map: NASA/STScI; ESO WFI; Magellan/U.Arizona/D.Clowe et al.
Clowe+2006 arXiv:0608407Bradac+2006 arXiv0608408Dark Matter
Hot and diffuse gas
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Lens equation:
Real position – source planeObserved position – lens plane
Lens equation
Angular diameter distances
The deflection angle depends on the “lens” mass distribution, and also on the observed position
For one position on the source plane , we can have multiple solutions for in some cases.
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Lens equation:
Real position – source planeObserved position – lens plane
Lens equation
Angular diameter distances
The deflection angle depends on the “lens” mass distribution, and also on the observed position
For one position on the source plane , we can have multiple solutions for in some cases.
12Deflection angle
From General Relativity and assuming a weak gravitational field (satisfied by galaxies, clusters and stars), the deflection angle of a “point mass” object is given by:
OBS: if you are observing regions very close to a black hole, this equation is not valid
For a extended lens we can “sum” the deflection angles of its mass density distribution ρ
Projected mass density profileWe are not considering perturbers along the line of sight
13Gravitational lens definitions
Dimensionless projected mass distribution, named convergence
Critical density
Lens potential , projection of the 3D gravitational potential
The important relations are:
14Magnification
In gravitational lensing, the surface brightness (= flux/solid angle) is conservedThe magnification is defined as the ration between the observed and intrinsic fluxes
For infinitesimal quantities, we can use the Jacobian matrix:
Information about the mapping between the source plane and lensing (observed) plane
Rewriting the lens equation as:
Short notation:
15Magnification
We can split this matrix in two parts, one isotropic and other traceless
We now define the shear , with
circular source mapping to thelens plane
Isotropic + magnification
shear
elliptical image
16Magnification
The total magnification is given by:
The two eigenvalues are related to the magnification in the tangential and radial directions:
TangentialRadial
RadialTangential
Example for an elliptical mass distribution
Source plane - Caustic Image plane – Critical lines
Demagnified central images
17Time delay
The time delay has two contributors, a geometrical and a gravitational
From the lens equation we have:
Therefore the lens equation can be rewritten as simply:
The time delay defines a two-dimensional surface and the multiple images are formed in the stationary points, i.e. minima, maxima or saddle points
Difference between the actual arrival time and the arrival time with no deflector
Credit: NASA / ESA
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Figure from Mollerach & Roulet, 2001, Gravitational Lensing And Microlensing
Example of a time delay surfaceThe point mass distribution is singular in the origin, the surface is not smooth, the central point diverges to infinity. Each source position has one surface.Time delay surface can be used to study the occurrence of multiple images
19Time delay
The time delay has two contributors, a geometrical and a gravitational
From the lens equation we have:
Therefore the lens equation can be rewritten as simply:
The time delay defines a two-dimensional surface and the multiple images are formed in the stationary points, i.e. minima, maxima or saddle points
Courbin, Saha and SchechterarXiv:0208043
Source plane Image plane
Time delay contours
Max.
min.
Difference between the actual arrival time and the arrival time with no deflector
20Time delayIf we measure the time delay between two images from the same source we can compute
F depends on the lens potential (mass distribution).Can be used to constraint the Hubble parameter
● see e.g. Suyu+2012● (chap. 1 and 2, arXiv:1208.6010)
HST colour compsite image
Tewes+2012
Light curve of the four multiple images
A
B
C
D
lensSuyu+2012 Time delay in galaxy
scale systems is few days – few hundred days (< year)
1 arcsec
21Caustics and critical lines
Source plane Time delay contoursImage plane
Einstein cross
Real image from Grillo+2018S1-4 are multiple images from one sourceThe bright galaxy in the acting as a gravitational lens
22Caustics and critical lines
Source close to a fold of the tangential caustic
Source plane Time delay contoursImage plane
Lens is elongated along the y-direction. Every time the source passes through a caustic, two extra images are formed
23Example of a fold system
From Oguri+2010 (arXiv:1005.3103)General relation for smooth mass distribution
Images B and C are outside the critical line● They have positive magnifications
A and D are inside the critical line● Negative magnifications
There is a central image E, with positive magnification but de-magnified
Usually difficult to observe because its low luminosity and contamination from the central galaxyOBS: the critical line is much more complicated
than this simple representation
24Caustics and critical lines
Source plane Time delay contoursImage plane
Source close to a cusp of the tangential caustic
Lens is elongated along the y-direction. Every time the source passes through a caustic, two extra images are formed
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HST colour compsite image
A
B
C
D
lens
1 arcsec
Example of a cusp system
General relation for smooth mass distribution
Images A and D are inside the critical line● They have negative magnifications
B and C are outside the critical line● Positive magnifications
No Central image is detected in this system
If we know well the mass distribution and have control on some systematics, we can use these relations to detect dark matter substructures
26Lens model 1: Point mass
Point mass:
This is an antisymmetric case, all vectors have the same direction, so we can reduce to one dimension. The lens equation is:
In this simple case we can solve for
Two different images from the same source position (the time delay surface is not smooth)
Deviations from the axial symmetry will “break” the ringFormally there is a central image, but this is an artefact because the point mass is singular
The magnification is given by:
27Microlensing
Point mass lenses are good approximations for starsMicrolensing refers to very small deflection angles (<0.01 arcsec) that we can not resolve, caused by compact objects on other sourcesObserved by monitoring stars during several days
Figure from Wambsganss+2006 (arXiv:0604278)
Caustics for of a point mass lens. The tangential caustic is degenerated in a central point
Trajectory of a background star with different impact parameters
28Detection of a planet using microlensing
Beaulieu+2006(arXiv:0601563)Perturbation caused by a object with ~5 times the Earth mass
First detected microlensing effect, Alcock+1993 (arXiv:9309052)
29Lens model 2: Axisymmetric mass distributionIn this case, the deflection angle depends on the mass enclosed inside the image position
Similar to the point mass lens, if we have an perfect alignment between observed, lens and source, we can compute:
If we have two different sources we can compute the projected mass within two radii, and measure the slope of the mass distribution!
For example:Mass of a lens at z=0.5 and source at z=2.0, forming an Einstein ring with 2 arcseconds
30Lens model 3
Singular Isothermal Sphere:Obtained by assuming that the matter distribution behaves as an ideal gas, thermal and hydrostatic equilibrium.
Navarro, Frenk and WhiteObtained from numerical simulations. The shape is universal regardless of the mass range
Central velocity dispersion
See for example Meneghetti notes for the calculation of the deflection angle and other quantities.
31Lenses with elliptical symmetry
We can construct projected mass distribution with elliptical symmetry by replacing the radial coordinate by an elliptical one:
Where:Ellipticity = 1 - b/a
Critical lines and causticsExample of a NFW model
In many mass models it is not possible to obtain analytical solution for the lens quantities, but we can solve numerically
Lenses with elliptical symmetry can produce most of the observed configurations of multiple images. But still very simplistic for real systems
Credit: M. Meneghetti
32Example of multi component lens
Sum of several mass components
Each galaxy member is described as a singular isothermal sphere, and the Dark Matter extended component by an elliptical mass distribution NFW or a cored IS
z cluster = 0.44; z sources = 1.4
33Weak lensing regimeRegions where the lensing effect weakly distorts the back ground galaxies
● Larger distances from the core
Assumes that background sources are randomly oriented → average out the intrinsic orientations of a sample of galaxies, obtaining the lensing signal.
Different systematics:● Observational seeing● Instrumental distortions● Intrinsic alignments, etc.
See Schneider+2005 (arXiv:0509252) for a review
Strong regime
34Weak lensing regime
Effect on the shape and sizes of background sources:
From wikipedia
You can measure the mass of galaxy clusters at large radiiStack the signal of several single galaxies to obtain an average profile
Projected mass distribution of a galaxy cluster
Umetsu+2015(arXiv:1503.01482)
Oguri+2010(arXiv:1004.4214)
35The real worldSystems with Einstein rings are rare → we use the multiple images or distortion to measure the mass distribution of galaxiesThe mass distribution is never axisymmetric → we have to consider at least elliptical symmetry and substructuresWe can model the mass distribution of cluster and galaxies using multiple images
Grillo+2015(arXiv:1407.7866)
36Probe the geometry of the universe
The angular diameter distances depend on the cosmological model
Flat Universe W = -1
Caminha+2016 arXiv:1512.04555
37Final remarks
Applications in many fields of Astronomy● Mass distribution of galaxies and galaxy clusters● Cosmological measurements, Hubble constant and geometry of the Universe● Distribution of mass substructures● Studies of faint and distant galaxies using systems with strong magnifications● etc
Mathematical formalism/theory (from General Relativity) is well established, but many effects are challenging to consider:
● Structures along the line of sight● Realistic mass distributions – higher order asymmetries (beyond elliptical symmetry)● Intrinsic alignments in weak lensing studies● etc
Perspectives:● Many observational programs using the best observatories● Hubble Space Telescope, ESO- Very Large Telescope● Approved programs with the James Webb Telescope to observe lensing systems● Euclid satellite will provide a very large sample (100x more than know today) of lenses
for robust statistical studeis
